Affine Bonnet surfaces

Abstract: The Bonnet problem in Euclidean surface theory is well-known: Given a metric $g$ and a function $H$ on an oriented surface $M^2$, when (and in how many ways) can $(M,g)$ be isometrically immersed in $\mathbb{R}^3$ with mean curvature $H$? For generic data $(g,H)$, such an immersion does not exist and, in the case that one does exist, it is unique up to ambient isometry. Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite-dimensional moduli space of $(g,H)$ for which the space of such "Bonnet immersions" has positive dimension.

The corresponding problem in affine theory (a favorite topic of Eugenio Calabi) is still not completely solved. After reviewing the results on the Euclidean problem by O. Bonnet, J. Radon, É. Cartan, A. Bobenko and others, I will give a report on affine analogs of those results. In particular, I will consider the classification of the data $(g,H)$ for which the space of "affine Bonnet immersions" has positive dimension, showing a surprising connection with integrable systems in the case of data $(g,H)$ for which the space of affine Bonnet immersions has the highest possible dimension.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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